1. Field of the Invention
The present invention pertains to methods for analyzing a signal that enable this signal to be analyzed by using an orthonormal base of functions called wavelets.
2. Description of the Prior Art
There is a very well known way to analyze signals by using either a Fourier series analyisis if this signal is periodic or a Fourier integral if this signal is not periodic. These methods are very powerful but have a number of drawbacks. These drawbacks are especially pronounced for short-duration signals for which these methods are clearly not suitable.
It is known, however, that the short-duration signals form a major part of natural phenomena, especially in the field of sound, where noises of all types rarely last more than some seconds and are often shorter than one second.
Some years ago, Jean MORLET proposed a method of analysis, called the method of analysis by wavelets, which is particularly suited to this field of short-duration signals and notably to pulse noises.
This analysis by wavelets is explained very comprehensively in the journal Pour La Science, September 1987. In particular, this document includes an explanation of the reasons for which the Fourier method is ill-suited to short-duration signals.
Given an orthonormal base of wavelets including including m+1 wavelets .PSI..sub.0 to .PSI..sub.m and a scale .PHI..sub.m, the analysis of a signal S(t) takes place in a series of operations shown schematically in FIG. 1: the signal S(t) sampled on N points is convoluted with the set of vectors of the orthonormal base .PSI..sub.0 to .PSI..sub.m and by .PHI..sub.m. The product of convolution of S(t) by .PSI..sub.0 gives N points determining the order 0 wavelet coefficients. The product of convolution of S(t) by .PSI..sub.1 gives N points which are the order 1 wavelet coefficients etc., up to the product of convolution of S(t) by .PSI..sub.m which gives the N order m wavelet coefficients. Finally, the product of convolution of S(t) by .PHI..sub.m gives N points which are the order m scale coefficients. It is noted that, to define the analyzed signal, which itself has N points, there are thus N points of analysis obtained. There is thus a redundancy of information which is, however, necessary in this method of analysis used notably with the so-called Haar wavelets.
Ingrid DAUBECHIES, of the AT&T Bell Laboratories at Murray Hill, has computed so-called compact support wavelets which are perfectly adapted to a fast analysis algorithm conceived by Stephane MALLAT of the University of Pennsylvania.
This algorithm, shown in FIG. 2, also starts from a signal S(t) sampled at N points, but only the wavelet .PSI..sub.0 and the scale .PHI..sub.0 are used in a series of p successive steps.
In the first step, a product of convolution of S(t) by .PSI..sub.0 and by .PHI..sub.0 is computed and one in every two of the N points of these products of convolution is selected: this gives twice N/2 points, that is, N points as at the outset. Clearly, this sub-sampling is done more easily by performing only one in two computations when computing the products of convolution.
The N/2 points coming from the product of convolution of S by .PSI..sub.0 are memorized, and in a second step, we compute the products of convolution by .PSI..sub.0 and by .PHI..sub.0 of the N/2 points coming from .PHI..sub.0 in the first step by selecting one out of two points of the N/2 points coming from the computation. This second step is therefore quite identical to the first one, except that the starting point is not the signal S(t) defined on N points but the product of convolution of S(t) by .PHI..sub.0 defined on N/2 points. The removal of one in every two points thus gives twice N/4 points.
For the following step, the N/4 points of the processing by .PSI..sub.0 are memorized and the same computation is started again on the N/4 points coming from the convolution by .PHI..sub.0. And so on.
At the step p, the method computes the products of convolution by .PSI..sub.0 and .PHI..sub.0 of the N/2.sup.p-1 points coming from the convolution by .PHI..sub.0 at the step p-1 by selecting one in two points, thus giving twice N/2.sup.p.
It is thus seen that, at this step p, there are twice N/2.sup.p points available, coming from this step p, plus the N/2, N/4 etc. N/2.sup.p-1 points memorized during the steps 1 to p-1, giving a total of N points that represent the analysis of the signal S(t) by the wavelet .PSI..sub.0 and the scale .PHI..sub.0. There is therefore no redundancy in the set of coefficients obtained, which have been shown by MALLAT to represent S(t) accurately.
This algorithm can continue, naturally, up to a step p such that 2.sup.p =N, but it has been observed experimentally that the coefficients thus obtained get stabilized very quickly at the end of a few steps, for example 5 to 6 steps. It is therefore possible to come to a stop very quickly in the successive computations, and this practice has been justified by proceeding to reconstruct the signal S(t) on the basis of the N points obtained at the step p, by doing the reverse processing operation. It is noted that the signal S(t) thus reconstituted is very similar to the initial signal once the coefficients are stabilized after 5 or 6 iterations as described further above.
When this analysis has been performed, what remains to be done is to use the N coefficients to obtain, for example, a classification of the analyzed signals. The problem is of the same nature as the one wherein, when the method of analysis into Fourier series is used, the origin of the analyzed signal is determined from the spectrum obtained. It is thus that the sound given out by a violin is distinguished from that given out by a saxophone, because these sounds do not have the same spectrum. It will be noted that these two instruments, taken as an example, can emit a continuous sound capable of being accurately analyzed by a Fourier series. In the case of a piano which gives out short notes, the use of a Fourier series is far less indicated and that of the analysis into wavelets would be far more worthwhile.
Different methods to interpret the coefficients of wavelets have been proposed. Thus, these coefficients may be introduced into a neuronal net which will perform the classification by a self-learning process. Another method consists in applying the signals to a display device XY. We thus obtain figures such as those shown in pages 35 and 36 of the article in POUR LA SCIENCE cited further above.
There is no way known, as yet, to identify these figures very well, and one of the problems that arises is that they take a tree shape which is relatively confused. This tree shape comes from the fact that, according to MALLAT's algorithms, we have a number of signals that decreases as and when the steps occur.